II.6 The Development of the Idea of Proof, Proof in the Twentieth Century: Pages 140 – 142.

In the twentieth century the actual mathematical proofs are seldom presented as fully formalized texts. They are arguments precise enough to convince a mathematician that it could be turned into a fully formalized text.

Hilbert and his collaborators around 1920 devised a notion of proof as a formalized and purely syntactic deductive argument. There soon came difficulties with this formalized proof idea. Around 1930 Godel came up with his incompleteness theorem. This showed that “mathematical truth” and “provability” are not the same thing. In any consistent sufficiently rich axiomatic system there are true mathematical statements that cannot be proved.

It was also proved that certain important mathematical statements are undecidable. In 1963 Paul Cohen established that the continuum hypothesis can be neither proved nor disproved in the usual axioms for set theory.

Some formal proofs are exceedingly long. The classification theorem for finite simple groups was worked out by large numbers of mathematicians in many parts and put together would reach about ten thousand pages. A proof so long that a single human being cannot check may be problematic to our conception of proof.

Sometimes the proofs are so complex that only a select few individuals are even capable of or qualified to evaluate the proof. Examples of this are the proof of Fermat’s last theorem and the Poincare conjecture. If someone claims to prove a theorem but no one wants to evaluate the proof then what is the status of the theorem?

With modern computers and probabilistic methods it is possible to prove theorems with a high probability of them being true. Are these proofs? Even though they may have less likely error than the long classification theorem for finite simple groups mentioned above!

In 1976 Kenneth Appel and Wolfgang Haken proved the four color theorem which required the use of a computer. This is an example of a computer assisted proof. The future may hold computer generated proofs.

Another issue is that some conjectures that appear to be true are the basis for much research under the assumption that they are true. How should the results of such research be viewed?

The formal notion of proof is as a string of symbols that obeys certain syntactical rules. This notion continues to provide an ideal model for the principles that underlie what most mathematicians see as an essence of their discipline.